Algebraic Reasoning

[Keith]

To what extent do these PLEs have the potential to develop algebraic
reasoning? Use specific characteristics of a PLE of your choice in
order to support your argument.

[Tenille]

I have chosen to use PLE 3: The sum of all numbers in the
multiplication table to demonstrate the algebraic reasoning potential
of the PLEs. First of all, I am unsure whether or not I should
include the construction of spreadsheet in this discussion. On the
one hand, the ability to reference a cell as a variable is one of the
great algebraic characteristics of a spreadsheet. On the other hand,
I felt that the syntax of constructing the spreadsheet in this case
interferred with exploring the spreadsheet itself (I spent a lot of
time trying to figure out how the formulas within the spreadsheet
worked). I also am unsure whether the author intended the teachers to
construct the spreadsheet themselves or use the already created
spreadsheet to explore the multiplication table. I think the later is
the more likely case. For those reasons I chose to discuss the
algebraic potential of PLE 3 with the condition that the spreadsheet
was already prepared for the teachers to use. Perhaps one of you guys
who is smarter than I am could explore the other case.

One important characteristic of algebraic reasoning is the recognition
and expression of relationships. I think that PLE 3 facilitated the
recognition of a relationship between the size of the multiplication
table and the sum of the numbers. This was done through the use of
the table. I initially found the pattern of squared triangular
numbers in the sum all the numbers column and then tried to figure out
how that related to the size of the multiplication table.
Specifically wondering, how I could create a function in terms of the
size of the multiplication table. I found the relationship between
the two variables and used various representations to display the
relationship (as a table, equation, and graph).

Displaying the data as a graph increased the potentential for
developing algebraic reasoning. Another aspect of algebraic reasoning
is studying the rate of change. While it is clear from the table that
the rate of change is not constant, the graph of the data provides a
greater opportunity to develop algebraic reasoning.

The final aspect of algebraic reasoning that I want to discuss in
regards to this PLE is that of generalization. This PLE faciliates
generalization in two ways: generalizing from the concrete to the
abstract case in the "inductive phase" and then generalizing from the
abstract case to all natural numbers in the "demonstrative phase".
Let me clarify the distiction I have made here. I think that being
able to generalize from the table of values to a possible equation
that works for all the values in the table is one form of
generalization that occurs in this PLE. Being able to then generalize
and show inductively that this equation will work for all natural
numbers is a different and more difficult form of generalization. I
think that the PLE did an exceptional job of facilitating this second
generalization by highlighting the gnomon when expanding from size 10
to size 11.

In summary, I think that this PLE helped develop algebraic reasoning
by allowing and promoting the exploration of relationships, the study
of change, and the generalization of their findings. I was intrigued
by all the mathematics "buried" within the multiplication table. It
reminded me of Pascal's triangle. Which leads me to my question that
was bothering me the entire time I read the article: Do you think that
preservice elementary teachers could really uncover all this
mathematics? I thought that some of the PLEs were akin to what we
solved in Problem Solving last semester as graduate students. I am
underestimating elementary teachers' abilities? I would be interested
to hear what you guys think.

[Shawn]

These PLEs have great potential to develop algebraic reasoning.
However, the teacher that would guide the students as they develop
their algebraic reasoning while investigating these tables needs to
have great insight into the mathematics discussed in this article. It
doesn't seem overly difficult, but it is tedious. (Perhaps this is
why they have such potential.) It is also difficult to not get bogged
down in the technology and the syntax of the formulas that any
mathematical reasoning that could be gained is lost.

The PLE that I have chosen to discuss the algebraic potential is PLE
2. Students can be directed to find a way to name every number in the
table by using one number from either the first row or column and a
number next to it, either above, below, right, or left. (Even though
this was not as far as the article went, I think that it would be a
good task.) They should come up with some ideas like were discussed
in this section of the article, but perhaps they might not know how to
formulate them into an algebraic notation. With a hint of say the
notation of P(x, y) = xy, they could generate the rest. (What would
be the possible solutions to this?) The boundary conditions could
also be brought up in a discussion. This is a great way to formalize
algebraic reasoning using a table they have known for years.

On Feb 20, 3:37 pm, Keith <LethaLeat...@gmail.com> wrote:

[Rachelle]

I think in general spreadsheets have great potential for developing
algebraic reasoning because of the opportunity to treat cells as
variables, write equations, generate sequences, etc. However, I'm not
sure I understand the potential these PLEs have for developing
algebraic reasoning. There are certainly great components of the PLEs
that can foster algebraic reasoning, but I'm not sure to what extent
the prospective teachers would really be able to access the
mathematics. I am excited to read others' blogs because I think your
insights will help me understand the PLEs better.

I think that these PLEs foster algebraic thinking through
opportunities to find patterns, write equations, and generalize
results. I will use PLE 1 to illustrate my point. If asked to format
the cells above and below a diagonal in different formats, students
would first need to examine what sets those two groups of numbers
apart. This could entail finding patterns in a specific (say 10x10)
multiplication table. Once students have identified a pattern, they
must attempt to write an equation that would describe the situation.
The ability to generate an equation from a given situation uses
algebraic reasoning. Then, as Abramovich discusses, students have to
generalize their conditional equations to work not only for the 10x10
table, but for a table of any size n. Through finding patterns,
writing equations, and generalizing results, students will most likely
further develop their algebraic thinking.

On Feb 20, 3:37 pm, Keith <LethaLeat...@gmail.com> wrote:

[Rachelle]

Shawn, I'm glad you brought up the syntax. I feel like I got bogged
down in the syntax and I was only reading the paper! I think that a
teacher would need to be very careful in striking a balance between
making sure the students are using their own reasoning to come up with
their formulas and making sure that their reasoning doesn't get lost
in the battle for the right syntax. How exactly would a teacher
present these PLEs to a class? Should the teacher give the spreadsheet
syntax whenever a student develops a formula they want to use? Or
would figuring out the right syntax help students further develop
algebraic reasoning?

> > order to support your argument.- Hide quoted text -
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> - Show quoted text -

[Janellie]

First of all, let me say how very surprised when I noted that these
PLEs and exercises were meant for prospective elementary teachers. I
am not sure that having these teachers learn all about proof by
induction is going to be all that beneficial for their algebraic
reasoning. I did not feel like the way the authors presented the
material was too effective, but I think that the dynamic spreadsheets
of the multiplication table would be very valuable. I think my main
issue with what was being asked of the elementary teachers was that it
had a lot of symbolic manipulation with the induction method and
seemed to focus less on mathematical sense-making and actual algebraic
reasoning.

That said, I think that some of these PLEs could be beneficial. I
discuss specifically PLE # 6. I like how this PLE allows students to
look at divisibility within the multiplication table. So, I think it
has GREAT potential in developing algebraic reasoning. First of all,
a class could talk about the rules of divisibility. What makes some
numbers divisible by 8, but not other numbers? Why are sometimes more
than 50% of the numbers divisible by 2? Even though I did not like
the extent of the formal mathematics discussed in this paper
(induction), I think elementary teachers should be required to
generalize patterns and formulas. What is the pattern? Why does that
pattern make sense? Students can reason about factors, multiples, and
divisibility.

The article also talks about the mental power that could be used
without the spreadsheet (see pg. 170). I agree with the authors when
they say it might be beneficial at certain times for students to
reason and make conjectures without using the technology.

Question: I am totally off base here? Help me understand why the
authors think these exercises would be good for a college student
preparing to be a elementary school teacher.

On Feb 20, 3:37 pm, Keith <LethaLeat...@gmail.com> wrote:

[erin...@byu.edu]

Let's be honest, I didn't love this article. I felt like the first
half was distracting because of the instructions to enter the programs/
formulas into a spreadsheet; once the more complex math hit,
Abramovich seemed to abandon these efforts, but then I became so lost
in trying to figure out what each PLE was supposed to do and what it
meant (not to mention the fact that some of them had problems working
as I was playing with them), that I had a very difficult time
following the mathematics that the teachers were supposed to be
exploring. Abramovich stated in the beginning of this paper that in
order for teachers "to be able to teach mathematics conceptually,
[they] themselves have to be taught in that way." Perhaps I'm
expecting an easier-to-follow curriculum than these PLEs, or perhaps I
was just frustrated with myself for not understanding something that I
expected to be easy to understand (if elementary school teachers can
do it without special math training, why can't I with all my
specialized expertise?!?) Anyways, I just needed to express my
opinion before I continued to address what we're supposed to be
addressing.

If I understood all of the PLEs as well as I did a few of them, I
think there is great potential to develop algebraic reasoning.
Throught my experiences mentoring in MathEd 306 I have come to expect
the unexpected in regards to the mathematical capabilities of these
prospective elementary school teachers. They are very capable of
creating formulas to represent many of the formulas discussed in this
paper. I wonder if the PLEs were presented to the elementary school
teachers with more of an explanation of what each one does and what
everything means, or at least a more concise explanation? If so, then
I believe that each of these are very capable of helping teachers
develop algebraic reasoning.

Take, for example, PLE 2: Assuming teachers are able to magically
notice the fact that "each number in the talbe...equals to the number
immediately to the left (or above) plus the number in the first column
of the table (or row)," then I certainly believe teachers will develop
a meaning for multiplication as repeated addition. I am at a loss for
how to help teachers discover this. I am skeptical that many of them
will see "more" than the traditional multiplication table; with all
their frills, buttons, colors, and sliders, I don't feel like the
spreadsheets will do such amazing things for the teachers (with the
exception of when we start discussing modulos, which I felt like were
never adequately explained: why do some of those tables only include
the mod number, and others only exclude the mod number columns/rows?
I digress.). Perhaps the students would be able to get great things
from it since they have never seen it before, but again, I wonder if
the teachers will be able to get past their old memorized uses of the
multiplication table to find the deeper math. Assuming that teachers
could discover this mythical connection, I believe they would
certainly be able to create some sort of equation(s) to express this
fact like unto the ones Abramovich discussed (pg 163). The ability to
create general terms that express patterns is certainly a vital part
of algebraic reasoning and certainly compliments/continues to build
teachers' reasoning skills and understanding.

As you can see, I'm skeptical. I see that there is potential in these
PLEs, but I am definitely not convinced. Maybe one of you can win a
blue ribbon for doing that? I'll go read what you said and announce
the winner tomorrow!

On Feb 20, 3:37 pm, Keith <LethaLeat...@gmail.com> wrote:

[CJ]

I would say that these PLEs definitely have potential for the
development of algebra as generalized arithmetic, centering on the
idea of using a variable as a generalized number to study
relationships between operations. I will use PLE 2 to help demonstrate
what I mean. This PLE has the idea of having students find different
ways to create the rows or columns of the multiplication table. It
suggests that students will realize that, although they are making a
table for multiplication, it is just as good to use repeated addition.
To make the 2's row, they can start with 2, add 2 to get 4 (2x2), add
2 more to get 6 (2X3) and so forth. So even though they did 2 + 2 + 2,
their result is 3x2 (or 2x3, but probably only on another continent).
After realizing that this pattern persists for each row of the table,
students could generalize the pattern using a variable to say a + a +
a = 3xa or 3a. I have recently observed that students have problems
really connecting addition and multiplication this way when using
variables, and some claim that "they just forgot," but I believe that
having a multiplication table in front of them and interpreting it in
terms of addition creates a powerful context for students to make this
connection on their own.
Abramahovich suggests going even further. He develops a "product
function" and write the product of x and y as P(x,y)=xy. A lot of
times when we introduce function notation, we introduce a myriad of
functions which are somehow all named f(x), do one example with each
function, and then complain when our students think that f(x) means f
times x. I like how Abramovich gives a context in which we use the
same function repeatedly with a purpose. This helps us to realize that
the notation really just is notation, or a different way of saying
that we want to multiply two numbers together. And then, we get into
the distributive property: P(x,y) = P(x-1,y)+y. Or, without the
notation, this means that xy=(x-1)y+y. Likewise, P(x+m,y)=P(x,y)
+P(m,y) means that (x+m)y=xy+my. Rather than being taught this as a
rule or pattern with symbols, the students can actually come up with
this on their own because the table gives them a lot of examples to
generalize from. This enforces the connection between arithmetic and
algebra, rather than introduce algebra as a whole different ballgame.

On Feb 20, 3:37 pm, Keith <LethaLeat...@gmail.com> wrote:

[CJ]

Hello everybody. I am replying to Janelle's post, but first I would
like to make a general comment as a reply to everyone's post and that
is that after reading a different article by the same author, I have
decided that he is very talented with spreadsheets and also talented
at uncovering and symbolically characterizing the mathematics in the
spreadsheet activities that he suggests. However, for some reason, he
never really gives an idea of what he thinks should actually be
happening in the classroom, or how real students would go about
uncovering these mathematics. I do believe that he thinks that
students should and can benefit a lot from creating their own
spreadsheets, but it is still unclear exactly how that happens. I
imagine his students are much better at managing spreadsheets than I
am. So, while I see great potential in his suggestions and ideas, the
actual implication seems to be left up to us.
Janelle, I love your questions. I can't say that I know what the
author think about a lot of things (the previous paragraph is an
example), but having spent some time on my own with pre-service
elementary school teachers, a lot of them seem to see math as a very
unconnected set of rules. Any activity that allows them to connect two
mathematical concepts in meaningful ways seems to be incredibly
motivating and empowering for them. I don't know why this is, but
perhaps it is a quality of mathematics that the people who choose to
study mathematics seem to recognize and appreciate. If mathematics
really was nothing but memorization, I would have had to give up on it
long ago. If pre-service teachers can change their attitudes about
mathematics (and technology, of course) by seeing that the
multiplication table is so much more than just memorization, then I
believe this is one step closer to changing how mathematics is taught.

[Janellie]

Thanks Chris, for the reply. As Erin said, I definitely think that
elementary teachers can do amazing things with mathematics that we
don't necessarily expect. I especially appreciate how Chris said that
Abramahovich is very knowledgeable about spreadsheets and has some
great ideas about teachers using these ideas. I would like to know
how he would go about helping students discover these things. I feel
like I have some skills in using the spreadsheet, but they are very
limited (example: A March to the North Pole). Chris also pointed out
some very good ways these PLEs could help students use algebraic
reasoning. She has experience teaching 306, so I believe her.

[CJ]

dry run (noun) (ca. 1942) 1: a practice firing without ammunition 2: a
practice exercise: REHEARSAL, TRIAL

On Feb 20, 3:37 pm, Keith <LethaLeat...@gmail.com> wrote:

[erin...@byu.edu]

Rachelle - the way you're speaking makes me think that you understand
that these prospective elementary school teachers are presenting these
PLEs to their elementary school classes. Is this correct? If we
pretend, instead, like the "teacher" you refer to is the instructor of
this preservice class and the "students" are the prospective teachers,
then I think the teacher should inquire what type of formula the
students want to create and what it does and then introduce them to
the basic syntax of creating a formula/program in the spreadsheet,
i.e., explain what "=if(___,___,___)" means in a spreadsheet or
"concatenate" or any of the other basic things. Then, with very
simple understanding of what tools are available, the students should
create their own programs/syntax/whatever you want to call it. I
think that it definitely helps them further develop algebraic
reasoning as they articulate exactly what they want to do and watch
others (the computer) carry it out. But, if we're not pretending, I
think that it would not be the greatest idea to try and have 3rd, 4th,
and 5th graders attempt to write complicated programs like those
discussed in the paper. There could be value (and thus I feel an
existence of potential) by having them create very simple, easy, small
programs because it would help develop their sense of argument/proof
and articulation of students' thought-processes is, I believe, helpful
to most everybody.

Janelle - I really enjoyed your question but think that Chris, with
her research, expressed things better than I ever could have, and so I
give kudos to her and defer to let her reign supreme on that throne.

> > - Show quoted text -- Hide quoted text -

[Tenille]

Chris, I really like how you brought up the connection between
arithmetic and algebra, especially with regards to the distributive
property. I think that a lot of teachers (elementary and secondary)
don't understand the importance of understanding the properties of
various operations nor do they understand the properties to begin
with. For example, the student teachers I interviewed for my thesis
could not explain how the distributive property related to
multiplication and addition, which is important to understanding how
the distributive property relates to adding like terms. I liked how
Abramahovich used the concreteness of arithmetic to begin getting
students to think about algebra. I think that is key to developing
algebraic reasoning (emphasis on developing). I think that the
traditional segregation of the two domains is one reason that the
student teacher I interviewed expressed the view that all whole
numbers are like terms (in a sense, she is right, but I think that
this indicates that she might not understand the connection between
algebra and arithmetic).

On Feb 28, 8:08 am, CJ <christine.johnson....@gmail.com> wrote: